On local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps

We study the 1:4 resonance for the conservative cubic Hénon maps C± with positive and negative cubic terms. These maps show up different bifurcation structures both for fixed points with eigenvalues ±i and for 4-periodic orbits. While for C–, the 1:4 resonance unfolding has the so-called Arnold dege...

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Bibliographic Details
Published in:Chaos (Woodbury, N.Y.) N.Y.), 2018-04, Vol.28 (4), p.043123-043123
Main Authors: Gonchenko, M., Gonchenko, S. V., Ovsyannikov, I., Vieiro, A.
Format: Article
Language:English
Subjects:
4-periodic orbits
Bifurcació, Teoria de la
Bifurcation theory
Bifurcations
Cubic Hénon map
Matemàtica
Matemàtiques i estadística
Mathematics
Strong 1:4 resonance
Àrees temàtiques de la UPC
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Summary:We study the 1:4 resonance for the conservative cubic Hénon maps C± with positive and negative cubic terms. These maps show up different bifurcation structures both for fixed points with eigenvalues ±i and for 4-periodic orbits. While for C–, the 1:4 resonance unfolding has the so-called Arnold degeneracy [the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient], the map C+ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by π/4. For both maps, several bifurcations are detected and illustrated.
ISSN:1054-1500
1089-7682
DOI:10.1063/1.5022764